Kontakt och symplektisk topologi med tillämpningar i fysik-inspirerad knutteori

Tidsperiod: 2017-01-01 till 2020-12-31

Projektledare: Tobias Ekholm

Finansiär: Vetenskapsrådet

Bidragstyp: Projektbidrag

Budget: 3 750 000 SEK

The research program studies contact and symplectic topology and its relations to other geometric fields, in particular to physics inspired low-dimensional topology and knot theory. The main tool is theories that use moduli spaces of holomorphic curves to extract contact and symplectic geometric information.From a physical perspective, well known and powerful polynomial knot invariants come from Chern-Simons gauge theory for 3-manifolds, which in turn is related to topological string theory in manifolds related to the cotanget bundle of the original 3-manifolds. Recent results relate the Gromov-Witten invariants of these theories with the simplest part of symplectic field theory, Legendrian contact homology, for the simplest holomorphic curves, disks, and the simplest 3-manifold, the 3-sphere. It is a main purpose of the research program to extend this relation to higher genus curves and to other 3-manifolds. Such an extension would both give a new way to approach these rich physical theories with solid mathematical tools and represent pioneering and guiding work for Lagrangian Floer theories deformed by higher genus curves. The program also intends to develop more internal aspects of contact and symplectic topology. In particular, the surgery perspective on the Fukaya category will be developed with applications to mirror symmetry in mind. Also, recent examples of rigid objects that behave as flexible from the point of holomorphic curve theories will be studied.